Here we present a combinatorial decision problem, inspired by the celebrated
quiz show called the countdown, that involves the computation of a given target
number T from a set of k randomly chosen integers along with a set of
arithmetic operations. We find that the probability of winning the game
evidences a threshold phenomenon that can be understood in the terms of an
algorithmic phase transition as a function of the set size k. Numerical
simulations show that such probability sharply transitions from zero to one at
some critical value of the control parameter, hence separating the algorithm's
parameter space in different phases. We also find that the system is maximally
efficient close to the critical point. We then derive analytical expressions
that match the numerical results for finite size and permit us to extrapolate
the behavior in the thermodynamic limit.Comment: Submitted for publicatio

Bartolo Luque

C. Moore

G. Hutton

J. P. Crutchfield

Lucas Lacasa

M. Gromov

M. Mezard

English

arXiv

We present a combinatorial decision problem, inspired by the celebrated quiz show called Countdown, that involves the computation of a given target number T from a set of k randomly chosen integers along with a set of arithmetic operations. We find that the probability of winning the game evidences a threshold phenomenon that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical simulations show that such probability sharply transitions from zero to one at some critical value of the control parameter, hence separating the algorithm's parameter space in different phases. We also find that the system is maximally efficient close to the critical point. We derive analytical expressions that match the numerical results for finite size and permit us to extrapolate the behavior in the thermodynamic limit

Lacasa Saiz de Arce, Lucas

Luque Serrano, Bartolome

Servicio de Coordinación de Bibliotecas de la Universidad Politécnica de Madrid

RAPID COMMUNICATIONS
PHYSICAL REVIEW E 86, 010105(R) (2012)
Phase transition in the countdown problem
Lucas Lacasa1,2,* and Bartolo Luque1
1Departamento de Matema´tica Aplicada y Estadı´stica, Escuela Te´cnica Superior de Ingenieros Aerona´uticos,
Universidad Polite´cnica de Madrid, 28040 Madrid, Spain
2Department of Physics, Clarendon Laboratory, University of Oxford, Oxford OX1 3PU, United Kingdom
(Received 30 May 2012; published 27 July 2012)
We present a combinatorial decision problem, inspired by the celebrated quiz show called Countdown, that
involves the computation of a given target number T from a set of k randomly chosen integers along with a set
of arithmetic operations. We ﬁnd that the probability of winning the game evidences a threshold phenomenon
that can be understood in the terms of an algorithmic phase transition as a function of the set size k. Numerical
simulations show that such probability sharply transitions from zero to one at some critical value of the control
parameter, hence separating the algorithm’s parameter space in different phases. We also ﬁnd that the system is
maximally efﬁcient close to the critical point. We derive analytical expressions that match the numerical results
for ﬁnite size and permit us to extrapolate the behavior in the thermodynamic limit.
DOI: 10.1103/PhysRevE.86.010105 PACS number(s): 05.70.Fh, 89.20.Ff, 89.75.−k
In combinatorial optimization problems, a large amount of
literature is directed at the occurrence of a so-called threshold
phenomenon in the performance of search algorithms [1–4]:
There exists a phase in parameter space where the search
algorithm can easily ﬁnd a solution to the aforementioned
combinatorial problem (as the number of available solutions
increases exponentially with the system size) and a phase
where such a solution typically (i.e., almost surely) does
not exist. The transition between both phases is sharp in
some situations, mimicking in several aspects the phenomenon
of a phase transition in statistical physics problems. Some
classical problems evidencing such a phenomenon include
combinatorial problems in random graphs or the satisfaction
of (random) Boolean clauses, generically gathered under the
umbrella of random constraint satisfaction problems [4,5].
Many of these concrete problems can indeed by interpreted
under a statistical physics formalism [6,7], the general idea
being that in a combinatorial optimization problem, in some
cases one can formalize a cost function to be minimized. In
satisfaction problems, this is, for instance, the number of
violated constraints. Within statistical physics of disorders
systems, such as in spin glasses, one indeed proceeds in the
same manner if the system is studied in the limit of low
temperature (in that situation, the system tries to adopt the
ground state or minimal energy conﬁguration). Thereby, the
cost function of a combinatorial optimization problem can
be related to the Hamiltonian of a disordered system at zero
temperature (for instance, ﬁnding a minimum partition within
the so-called partitioning problem is equivalent to ﬁnding
the ground state of an inﬁnite-range Ising spin glass with
Mattis-like, antiferromagnetic couplings [7]).
In this work we present a random decision problem, called
the countdown problem, which is inspired by a celebrated
British television quiz show called Countdown (based on the
French game show Des Chiffres et des Lettres, one of the
longest-running game shows in the world, and called other
names in several countries [8]). This show consists of several
*lucas.lacasa@upm.es
games, one of which incorporates the combinatorial problem
of arithmetically combining several numbers to produce
another one. Concretely, the contestants must use arithmetics
to reach a given target number from six other numbers, using
each of them at most once. Here we formalize a random
version of this decision problem and explore its solvability as
a function of the parameter space. We will provide numerical
evidence according to which the solvability of the decision
problem shows the presence of an algorithmic phase transition
and introduce an approximate theory, which we show to be
in good agreement with the numerics for ﬁnite-size systems
and permits us to theoretically ﬁnd the phase transition in the
thermodynamic limit.
Let us begin by deﬁning the pool of size M as the
integer interval [1,M]. Suppose that we randomly extract
with reposition from this pool a set of k integers X =
{x1,x2, . . . ,xk} and another integer T called the target. Stated
as a decision problem, the target game raises the following
question: For a given duple (k,M), what is the probability
P (k,M) of reaching T by combining the elements ofX (where
each element xi can be used or not, but each of them will be
used at most once) through the set of arithmetic operations
A = {+, −, ×,÷}? Once M is ﬁxed, one can assert that for
rather small values of k, such as k = 2, the number of possible
combinations is rather limited. For large values of k, the
situation is the opposite: On average there will exist many
possible ways of combining the elements in X to reach T .
Whereas several parallels with random k satisﬁability can
be outlined, it can be shown that the problem at hand is
not based on ﬁnding the correct variables assignment X , but
the correct Hamiltonian assignment H among a Hamiltonian
ensemble [9]. While this dual representation precludes the
possibility of a standard statistical mechanics approach, we
will take advantage of the number-theoretic nature of the
problem to propose a probabilistic-theoretic treatment. We
start be relaxing the problem statement by assuming that the set
of available arithmetic operations is restricted to A = {+,−}.
As the problem is computationally nontrivial [10], in order
to explore the problem numerically, we have implemented a
brute force recursive routine that explores the search space in
010105-11539-3755/2012/86(1)/010105(4) ©2012 American Physical Society
RAPID COMMUNICATIONS
LUCAS LACASA AND BARTOLO LUQUE PHYSICAL REVIEW E 86, 010105(R) (2012)
k
1 2 3 4 5 6 7 8 9 10 11
0
0.2
0.4
0.6
0.8
1
P(k,M)
M = 29
M = 28
M = 27
0
8
20 1500
c
M
k
FIG. 1. (Color online) Winning probability P (k,M) curve as a
function of k, for increasing pool sizes M , in the system with
A = {+,−}. Solid curves are the analytical predictions [Eqs. (2)–(4).
The inset shows a linear-logarithmic plot of the crossover value
scaling kc(M) for different values of the pool size M . In each case
the crossover value is estimated as the linear interpolation of k for
which P (kc,M) = 0.5, as in percolation theory. The straight line is a
ﬁt to a logarithmic function kc(M) = a ln(M) + b with a = 0.98 and
b = 0.31.
an exhaustive way (additional details are given in Ref. [9]). For
a given pool size M , we ﬁx k, make Monte Carlo simulations,
and perform ensemble averages over different realizations of
T and X . The winning probability P (k,M) is deﬁned as the
probability of reaching T by arithmetically combining at most
the k numbers (where each of the k elements can be used or not,
but each of themwill be used atmost once). In Fig. 1we plot the
results of P (k,M) vs k for different pool sizes M , averaged
in each case over 104 realizations. Note that the transition
between P ∼ 0 (losing almost surely) and P ∼ 1 (winning
almost surely) is sharp and occurs at a certain kc(M), estimated
as the linear interpolation of k for which P (kc,M) = 0.5, as
usual in percolation theory. This crossover value depends on
the pool size since the control parameter k is not intensive. In
the inset of Fig. 1 we plot, on a linear-logarithmic scale, its
dependence on system size, ﬁnding a logarithmic scaling of
the form kc(M) = a ln(M) + b, with a = 0.98 and b = 0.31.
Our analytical treatment deals with the estimation of
P (k,M), where k  M , and for that task we begin by
considering a single operation, the sum. Ifwe choose at random
n numbers from [1,M], the largest possible value of its sum is
nM . In a ﬁrst approximation, we will suppose that the sum of n
numbers is uniformly distributed in [1,nM]. The result of this
sum will fall in [1,M] with probability 1/n. The number of
results in [1,M] accessible from k numbers chosen at random
from that interval is therefore
∑k
n=1
1
n
( kn ) (note at this point
that arguments involving the central limit theorem are not
employed here as long as we are dealing with partial sums
of 2,3, . . . ,k random variables instead of simple sums of k
random variables and in the general case of more arithmetic
operations the theorem does not apply).
Let us proceed by introducing a second operation in our
system: substraction. Incorporating this operation is analogous
to handling the former system, where now each number a from
the n numbers chosen at random represents both a and −a.
Now the result of summing up the 2n numbers will belong to
[−nM,nM], such that, assuming uniformity once again, it will
fall in [1,M] with probability 1/2n. Proceeding as before, the
number of results in [1,M] accessible from k numbers chosen
at random from that interval N (k) can be written as
N (k) =
2k∑
n=1
1
2n
(
2k
n
)
≈ 2
2k − k − 2
2k + 1 . (1)
The third approximation consists in assuming that those N
by-products are indeed random independent trials of ﬁnding
the target in [1,M]. In that situation, the winning probability
P (k,M) reads
P (k,M) = 1 −
(
1 − 1
M
)N(k)
≈ 1 − e−N(k)/M (2)
for M  1. Although this result is based on crude approxi-
mations assuming independence and uniformity [which yields
a value for N (k) that is independent of M], it still works
qualitatively (results not shown). In order to gain quantitative
accuracy, we can take Eq. (1) up to leading order and as an
ansatz we introduce a dependence on M through a modulating
factor r(M) that quantiﬁes the correlations among numbers
(i.e., the lack of independence) such that
N (k,M) = e
r(M)k
k
. (3)
In order to estimate r(M), note that kc(M) is deﬁned such that
P (kc,M) = 1/2. From Eq. (2) we ﬁnd
r(M) = ln(Mkc ln 2)
kc
. (4)
In Fig. 1 we plot, as solid lines, the theoretical values of
P (k,M) as a result of Eqs. (2)–(4), with the appropriate
scaling kc(M) previously reported: The ansatz yields a self-
consistent result provided kc(M) is a logarithmic function.
In Fig. 2 we extend the problem to the more general case
in which all elementary arithmetic operations are allowed,
A = {+, −, ×,÷}. The numerical results are analogous to
the simpler case, where now the ﬁnite-size scaling of the
critical point fulﬁlls kc(M) = a lnM + b, with a = 0.84 and
b = 0.39. Solid lines are the predictions of the theory with
the latter scaling, showing again good agreement with the
numerical simulations.
Introducing the intensive control parameter α = k/kc, in
Fig. 3 we plot the numerical curves P (α,M) resulting from
the simulations performed in the general case. As the system
size increases, the probability of satisfying the game gets
sharper around the now-size-independent crossover value. The
behavior in the thermodynamic limit (M → ∞, α ﬁnite) can
be derived from Eqs. (2)–(4), ﬁnding a Heaviside step function
P∞(α) = lim
M→∞
1 − e−er(M)αkc /αkcM =
{
0 for α < 1
1 for α > 1, (5)
i.e., the onset of a true phase transition.
Finally, let us deﬁne the functionQ(k,M) that measures the
system’s efﬁciency as the average amount of potential targets
010105-2
RAPID COMMUNICATIONS
PHASE TRANSITION IN THE COUNTDOWN PROBLEM PHYSICAL REVIEW E 86, 010105(R) (2012)
k
1 2 3 4 5 6 7 8 9 10
0
0.2
0.4
0.6
0.8
1
P(k,M)
M = 27
M = 210
M = 29
M = 28
0
8
20 1500
c
M
k
FIG. 2. (Color online) Numerical results similar to those in Fig. 1
in the general case of A = {+, −, ×,÷}. The scaling of the critical
point is again logarithmic kc(M) = a ln(M) + b, with a = 0.84 and
b = 0.39, which shifts the critical point towards smaller values of k.
Solid lines are the results of the theory [Eqs. (2)–(4)] with the latter
scaling relation.
in [1,M] that can be reached per unit number k. This function
can be written as
Q(k,M) = P (k,M)M
k
, (6)
whose behavior is shown in Fig. 4. This measure reaches
a maximum in a neighborhood of kc(M) such that in the
0
8
20 1500
c
M
k
α
0 0.5 1 1.5 2
0
0.2
0.4
0.6
0.8
1
P( ,M)
M = 26
M = 210
M = 29
M = 28
M = 27
α 0.5
FIG. 3. (Color online) Winning probability P (α,M) curve as a
function of the intensive control parameter α = k/kc(M), where
kc(M) = a ln(M) + b is a scaling function of the crossover value with
size (the ﬁtting values are a = 0.84 and b = 0.39) according to Fig. 2,
for increasing system sizes. The curves get sharper as M increases,
indicating a phase transition in the thermodynamic limit: the onset of
the so-called threshold phenomenon. TheHeaviside function, reached
only in the thermodynamic limit, is a result of the theory (see the text).
0
8
20 1500
c
M
k
k
1 2 3 4 5 6 7 8 9 100
20
40
60
80
100
120
140
Q(k,M)
M = 27
M = 210
M = 29
M = 28
FIG. 4. (Color online) Efﬁciency measure Q(k,M) curve as a
function of k, for increasing pool sizes M , in the system with A =
{+, −, ×,÷}. Solid curves are the analytical predictions [Eq. (6)].
This measure is maximized near the transition point.
thermodynamic limit it diverges for α → 1, deepending on the
conjecture that states that the complexity of multicomponent
systems is maximized close to their critical points [11,12].
This behavior is also related to the so-called easy-hard-easy
pattern that takes place in algorithmic phase transitions and
suggests that close to the critical point (hard phase) the
computation time that the algorithm needs to come up with
a solution is maximized: For α < 1 the algorithm easily
ﬁnds that the problem is unsatisﬁable, whereas for α > 1 the
algorithm easily ﬁnds one of many solutions; when α ∼ 1
the number of solutions per unit number is optimal and the
algorithm needs to perform an exhaustive search of the whole
space to ﬁnd it. Note that within the quiz show Countdown,
the pool is bounded to M = 1000. Interestingly enough,
the contestants are allowed to make use of k = 6 numbers,
which corresponds, according to our previous analysis, to the
threshold between almost surely unsolvable to almost surely
solvable instances. Driving the system towards the critical
point ensures that the game is hard but typically solvable,
that is, interesting.
To summarize, in this work we have presented a random
combinatorial decision problem that evidences an algorithmic
phase transition separating the parameter space where the
problem is either almost surely satisﬁable or unsatisﬁable.
This work depends on the relationship between number
theory and theoretical physics, whose interface has been
shown to potentially promote [13] new approaches in both
areas.
Several remarks can be made. Note that, while in physical
phase transitions the ﬁnite-size scaling is usually in the form of
a power law shape kc(M) ∼ Ma with some ﬁnite size exponent
a, in this problem we ﬁnd logarithmic scaling kc(M) =
a ln(M) + b (insets of Figs. 1 and 2). From a thermodynamic
viewpoint, the logarithmic scaling is not problematic since it
lacks an a priori physical (i.e., thermodynamic) interpretation.
If the reader is uneasy, we emphasize that in order to
010105-3
RAPID COMMUNICATIONS
LUCAS LACASA AND BARTOLO LUQUE PHYSICAL REVIEW E 86, 010105(R) (2012)
build a thermodynamic formalism we could always deﬁne an
alternative control parameter ˜k ≡ exp(k) in order to recover
the usual power law scaling with system size ˜kc(M) ∼ Ma
and make this parameter intensive (a temperature) through
α˜ = ˜k/Ma . In contrast, we note that logarithmic scalings
have been found previously in other number-theoretic systems
evidencing collective phenomena [14–16].
Finally, while the transition between unsatisﬁable and
satisﬁable phases (losing and winning) occurs at lower values
in the general case A = {+, −, ×,÷} than in the simple
one A = {+,−}, in both situations the system evidences
the threshold phenomenon. Is this phenomenon restricted to
only elementary arithmetic systems or, to the contrary, is
this a fundamental behavior in abstract algebraic structures
deﬁned as a set of elements with some binary operations?
In this respect, note that the logarithmic scaling kc(M) can
be interpreted as the growth rate of the minimal number
of elements needed to cover a growing system through
binary operations since the number of possible outcomes of
combined binary operations grows exponentially fast. This is
a challenging open question for future research, which could
be addressed within combinatorial group theory [17].
We thank Re´mi Monasson for pointing out to us the
relation of this problem to combinatorial group theory. We
acknowledge ﬁnancial support from the programMODELICO
andComunidad deMadrid throughGrant No. FIS2009-13690.
L.L. thanks the Systems and Signals Group at the University
of Oxford, where part of this research was developed. The
comments of two anonymous referees are also acknowledged.
[1] S. Kirkpatrick and B. Selman, Science 264, 1297 (1994).
[2] R. Monasson, R. Zecchina, S. Kirkpatrick, B. Selman, and
L. Troyansky, Nature (London) 400, 133 (1999).
[3] D. Achlioptas, A. Naor, and Y. Peres, Nature (London) 435, 759
(2005).
[4] C. Moore and S. Mertens, The Nature of Computation (Oxford
University Press, Oxford, 2011).
[5] Handbook of Satisfiability, edited by A. Biere, M. Heule, H. van
Maaren, and T. Walsch (IOS, Amsterdam, 2009).
[6] M. Mezard, G. Parisi, and M. Virasoro, Spin Glass Theory and
Beyond (World Scientiﬁc, Singapore, 1987).
[7] S. Mertens, Theor. Comput. Sci. 265, 79 (2001).
[8] Wikipedia entry for Countdown game show:
http://en.wikipedia.org/wiki/Countdown_(game_show).
[9] L. Lacasa, and B. Luque (unpublished).
[10] G. Hutton, J. Funct. Program. 12, 6 (2002).
[11] J. P. Crutchﬁeld and K. Young, in Complexity, Entropy, and the
Physics of Information, edited byW.H. Zurek (Addison-Wesley,
Reading, MA, 1990).
[12] R. V. Sole´, S. C. Manrubia, B. Luque, J. Delgado, and
J. Bascompte, Complexity 2, 49 (1996).
[13] Number theory and physics repository: http://
empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/physics.htm.
[14] B. Luque, L. Lacasa, and O. Miramontes, Phys. Rev. E 76,
010103(R) (2007).
[15] L. Lacasa, B. Luque, and O. Miramontes, New J. Phys. 10,
023009 (2008).
[16] B. Luque, O. Miramontes, and L. Lacasa, Phys. Rev. Lett. 101,
158702 (2008).
[17] M. Gromov, in Geometric Group Theory, edited by
G. Niblo and M. Roller (Cambridge University Press,
Cambridge, 1993).
010105-4


Phase transition in the Countdown problem

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